Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.rs
Appl. Anal. Discrete Math. 11 (2017), 123–135.
DOI: https://doi.org/10.2298/AADM1701123A
A NOTE ON THE NORDHAUS-GADDUM TYPE
INEQUALITY TO THE SECOND LARGEST
EIGENVALUE OF A GRAPH
Nair Abreu, André E. Brondani∗ , Leonardo de Lima, Carla Oliveira
In honour of Dragoš Cvetković on the occasion of his 75th birthday.
Let G be a graph on n vertices and G its complement. In this paper, we
prove a Nordhaus-Gaddum type inequality to the second largest eigenvalue
of a graph G, λ2 (G),
r
n2
λ2 (G) + λ2 (G) ≤ −1 +
− n + 1,
2
when G is a graph with girth at least 5. Also, we show that the bound above
is tight. Besides, we prove that this result holds for some classes of connected
graphs such as trees, k−cyclic, regular bipartite and complete multipartite
graphs. Based on these facts, we conjecture that our result holds to any
graph.
1. INTRODUCTION
Let G = (V, E) be a simple undirected graph with n vertices and m edges
and let G be its complement. Given a vertex v ∈ V, the set of neighbors of v is
N (v) = {w ∈ V |{v, w} ∈ E} and d(v) = |N (v)| is the degree of v. The girth of
G is the length of the shortest cycle in G. Write A(G) = [aij ], where aij = 1 if
{i, j} ∈ E(G) and aij = 0, if {i, j} ∈
/ E(G), for the adjacency matrix of G. The
∗
Corresponding author. André E. Brondani
2010 Mathematics Subject Classification. 05C50.
Keywords and Phrases. adjacency matrix, second largest eigenvalue of a graph,
Nordhaus-Gaddum inequalities.
123
124
N. Abreu, A. E. Brondani, L. de Lima, C. Oliveira
characteristic polynomial of G is pG (λ) = det(λI − A) such that the roots λi for
i = 1, · · · , n are the eigenvalues
of A(G) and ocan be arranged as λ1 ≥ · · · ≥ λn .
n
(s )
(s )
(s )
The multiset Spec(G) = λ1 1 , λ2 2 , . . . , λt t , where si is the multiplicity of λi ,
1 ≤ i ≤ t, is the spectrum of the graph G.
In 2007, Nikiforov [17] proposed the study of the Nordhaus-Gaddum type
inequalities for all eigenvalues of a graph defining a function given by
(1)
max |λk (G)| + |λk (G)|
|G|=n
for all k = 1, . . . , n. For the case k = 1, previous papers addressed the problem
of finding lower and upper bounds to (1) as one can cite Nosal [19], Amin and
Hakimi [1], Csikvári [4] and Terpai [27]. The latter obtained the following sharp
upper bound:
4
λ1 (G) + λ1 (G) ≤ n − 1.
3
The general case, for any k, was first introduced by Nikiforov in [17]. Particularly,
in the case k = 2, Nikiforov and Yuan [18] obtained the best known upper bound
to (1) as
(2)
n
λ2 (G) + λ2 (G) ≤ −1 + √ .
2
In this paper, we propose a slightly improvement of inequality (2), that is,
r
(3)
λ2 (G) + λ2 (G) ≤ −1 +
n2
− n + 1,
2
what we will refer throughout the paper by N G2 -bound. Also, we show that N G2 bound holds for some classes of graphs such as trees, k−cyclic, regular bipartite,
complete multipartite graphs, generalized line graphs and exceptional graphs. For
the first two classes, the proofs appear in Section 2 while the others are in Section 3.
Also, in Section 3, we state our main result proving that inequality (3) is true for
all graphs of girth at least 5. In addition, a family of graphs presented by Nikiforov
in [17] is showed to be extremal to our N G2 -bound in Section 4.
2. PRELIMINARY RESULTS
In this section, we prove that inequality (3) holds for trees and k−cyclic
graphs when k ≥ 0. Some useful results are revisited to open this section. The first
one is the well-known Weyl’s inequality and the second one is an upper bound to
the largest eigenvalue of trees.
For convenience, given a Hermitian matrix M of order n, we index its eigenvalues as α1 (M ) ≥ · · · ≥ αn (M ).
A note on the Nordhaus-Gaddum type inequality to . . .
125
Theorem 1 ([13]). Let M and N be Hermitian matrices of order n, and let
1 ≤ i ≤ n and 1 ≤ j ≤ n. Then
αi (M ) + αj (N ) ≤ αi+j−n (M + N ) if i + j ≥ n + 1,
αi (M ) + αj (N ) ≥
i + j ≤ n + 1.
√
Theorem 2 ([3]). If T is a tree of order n then λ1 (T ) ≤ n − 1. The equality
holds if and only if T ' Sn , where Sn is the star with n vertices.
αi+j−1 (M + N )
if
Both results, Theorems 1 and 2, are used to prove in Proposition 3 that the
proposed N G2 -bound holds for all trees.
Proposition 3. If T is a tree of order n, then
r
(4)
λ2 (T ) + λ2 (T ) ≤ −1 +
n2
− n + 1.
2
Equality holds if and only if T ' P4 .
Proof. From computational experiments, it is easy to check that inequality (4)
follows for all trees up to 6 vertices and among them the
graph P4 is the only one
√
for which the equality holds since λ2 (P4 ) = λ2 (P4 ) = 5−1
2 .
Now, suppose that n ≥ 7. From Theorem 1,
λ2 (T ) ≤ −1 − λn (T ).
Since λn (T ) = −λ1 (T ), from Theorem 2, we get
√
(5)
λ2 (T ) ≤ −1 + n − 1.
(6)
(7)
From [22] [see Theorem 1], it is known that
r
n
λ2 (T ) ≤
− 1.
2
Using inequalities (5) and (6), we obtain,
r
√
n
λ2 (T ) + λ2 (T ) ≤ −1 +
−1+ n−1
2
r √
n
≤ −1 + 2 n − 1 + − 1 = −1 + 3n − 4.
2
Note that, for n ≥ 7,
3n − 4
(8)
=
<
2
n2
n
−n+1 −
− 4n + 5
2
2
n2
− n + 1.
2
So, by (7) and (8), the result follows.
126
N. Abreu, A. E. Brondani, L. de Lima, C. Oliveira
Proposition 4 relates the spectral radius of a graph to N G2 -bound.
√
n2
Proposition 4. Let G be a graph of order n, let x =
− n + 1 and y = −1 + x.
2
√
x
If λ1 (G) ≤
, then λ2 (G) + λ2 (G) ≤ y.
2
√
Proof. Let G be a graph of order n. Suppose that λ1 ≤ x/2. By applying Theorem 1 for the adjacency matrices of G and G, we obtain λ2 (G) + λn (G) ≤ −1.
Since λn (G) < 0 and |λn (G)| ≤ λ1 (G), we get
λ2 (G) ≤
−1 + λ1 (G),
and then
r
λ2 (G) + λ2 (G) ≤ −1 + 2λ1 (G) ≤ −1 +
n2
− n + 1 = y.
2
Now, let Sn+ be the star graph plus one edge. For n ≥ 9, the next result
shows that Sn+ has the largest spectral radius among all unicyclic graphs. By
Proposition 5 with simple algebraic manipulations, we show that inequality (3) is
true for all unicyclic graphs.
Proposition 5 ([23]). If G is unicyclic of order n, then
2 ≤ λ1 (G) ≤ λ1 (Sn+ ).
√
Moreover, for every n ≥ 9, λ1 (Sn+ ) ≤ n.
Proposition 6. If G is unicyclic of order n, then
r
n2
λ2 (G) + λ2 (G) ≤ −1 +
− n + 1.
2
Proof. From computational experiments, it is easy to check that the result follows
for all unicyclic graphs up to 9 vertices. Now, suppose that G is an unicyclic graph
with n ≥ 10. Note that,
(9)
4n
<
n2
− n + 1.
2
From inequality (9) and Proposition 5, we have
√
n
r
1 n2
<
− n + 1.
2
2
λ1 (G) ≤
By Proposition 4, we obtain the desired inequality.
A note on the Nordhaus-Gaddum type inequality to . . .
127
For a non-negative integer k, a connected graph of order n and size m is
k−cyclic if m = n + k. For k = 0, the result folllows from Proposition 6. For each
k ≥ 1, we give the proof in Proposition 7 under some condition on n.
Proposition
7. Let k be a positive integer. If G is a k−cyclic graph of order
√
n ≥ 5 + 16k + 31, then
r
n2
λ2 (G) + λ2 (G) ≤ −1 +
− n + 1.
2
Proof. From Hong [12],
λ1 (G) ≤
(10)
For each n ≥ 5 +
√
n + 2k + 1.
16k + 31, we get
4(n + 2k + 1)
(11)
√
=
≤
n2
n2
− n + 1 + (−
+ 5n + 8k + 3)
2
2
n2
− n + 1.
2
Using equations (10) and (11),
1
λ1 (G) ≤
2
r
n2
− n + 1,
2
√
for all n ≥ 5 + 16k + 31.
From Proposition 4, the result follows.
This section is ended by two results that will be useful to prove our main
results.
Proposition 8 ([16]). For a graph G of order n and girth at least 5,
√
λ1 (G) ≤ min{∆, n − 1}.
Proposition 9 ([24]). For n ≥ 2, if G 6' Kn is a graph of order n then λ2 (G) ≥ 0.
The equality holds if and only if G is a complete multipartite graph.
3. MAIN RESULTS
For a natural number p, 1 ≤ p ≤ n, a graph G on n vertices is a complete split
graph, denoted by CS(n, p), if it can be partitioned into an independent set of p
vertices and a clique of n − p vertices such that every vertex in the independent set
is connected to every vertex of the clique. Notice that CS(n, p) is isomorphic to
128
N. Abreu, A. E. Brondani, L. de Lima, C. Oliveira
the complete multipartite graph Ks1 ,s2 ,··· ,sn−p+1 such that si = 1 for 1 ≤ i ≤ n − p
and sn−p+1 = p.
Next, we prove that the complete graph on n vertices is the only one graph
which attains λ2 (Kn ) + λ2 (Kn ) = −1. So, assuming that G is not isomorphic to
Kn , it turns out that the lower bound to λ2 (G) + λ2 (G) is obtained for a complete
split graph CS(n, p).
Proposition 10. Let G be a graph with order n ≥ 3 with at least one edge. Then,
one of the following statements holds,
(i) If G 6' Kn then λ2 (G) + λ2 (G) ≥ 0. Moreover, if G has no isolated vertices,
λ2 (G) + λ2 (G) = 0 if and only if G is a complete split graph;
(ii) λ2 (G) + λ2 (G) = −1 if and only if G ' Kn ;
(iii) There is no graph G such that λ2 (G) + λ2 (G) ∈ (−1, 0).
Proof. Let G be a graph of order n ≥ 3 with at least one edge.
(i) Let G 6' Kn . By Proposition 9, λ2 (G) ≥ 0. Besides, since G 6' nK1 and
G 6' Kn then
λ2 (G) + λ2 (G) ≥ 0.
For an integer p ≥ 2, let G ' CS(n, p). Since CS(n, p) ' K1,...,1,p , from
Proposition 9, we get λ2 (G) = 0. It is known that PG (x) = xn−p (x+1)p−1 (x−
p + 1). Then λ2 (G) = 0 and so, λ2 (G) + λ2 (G) = 0.
Now suppose that λ2 (G) + λ2 (G) = 0. Hence, G 6' Kn and, from Proposition 9, λ2 (G) = λ2 (G) = 0. Also, G ' Kp1 ,...,pk , for positive integers
1 ≤ p1 ≤ · · · ≤ pk such that 1 < pk .
Consequently, G ' Kp1 ∪· · ·∪Kpk and λ2 (G) = pk−1 −1. However, λ2 (G) = 0.
Then, pk−1 = 1 and so, p1 = p2 = · · · = pk−1 = 1.
(ii) If G ' Kn , λ2 (G) = −1 and λ2 (G) = 0. Reciprocally, if λ2 (G) + λ2 (G) = −1
then λ2 (G) < 0 or λ2 (G) < 0. From Proposition 9, we have G ' Kn .
(iii) The proof of this case follows straightforward from (i) and (ii).
Proposition 11. Let G be a graph of order n and girth at least 5. Then,
r
n2
− n + 1.
λ2 (G) + λ2 (G) ≤ −1 +
2
Proof. The proof will be done in two cases.
Case 1: Let G be a graph of order n and girth g such that n, g ∈ [5, 8].
A note on the Nordhaus-Gaddum type inequality to . . .
129
Under such conditions, there exist 26 graphs and all of them are unicyclic
(see in [21]). From Proposition 6, we get
r
n2
λ2 (G) + λ2 (G) ≤ −1 +
− n + 1.
2
Case 2: Let G be a graph of order n ≥ 9 and girth g ≥ 5. Note that
2
2
n
n
−n+1 −
− 5n + 5
4(n − 1) =
2
2
√
√
n2
1
=
− n + 1 − (n − 5 + 15)(n − 5 − 15)
2
2
n2
(12)
− n + 1.
<
2
Theorem 8 and inequality (12) give us
λ1 (G) ≤
<
√
n−1
r
1 n2
− n + 1.
2
2
From Proposition 4,
r
λ2 (G) + λ2 (G) ≤ −1 +
n2
− n + 1.
2
For regular graphs, it seems that N G2 -bound is true and some insights
from [7] can be useful in order to prove it. Here, we prove that inequality (3)
holds for every connected regular bipartite graph.
Proposition 12. If G is a connected r-regular bipartite graph of order n, then
r
n2
λ2 (G) + λ2 (G) ≤ −1 +
− n + 1.
2
Proof. For a r-regular graph G, it is known [see Theorem 3.1 of [26]] that
λ2 (G) ≤
(13)
n
− r.
2
From Theorem 1 and since G is a r-regular bipartite graph,
λ2 (G) = −1 − λn (G) = r − 1.
However, from (13),
(14)
λ2 (G) + λ2 (G) ≤ −1 +
n
.
2
130
N. Abreu, A. E. Brondani, L. de Lima, C. Oliveira
Notice that for n ∈ R,
n
≤
2
(15)
r
n2
− n + 1.
2
Hence, from inequalities (14) and (15), the proof is completed.
Graphs with the property λ2 ≤ 1 are investigated in several papers and they
are completely described within certain families (see for exemple [5], [8], [9], [14],
[15], [20] and [25]). The next result shows that all graphs with the mentioned
property satisfy N G2 -bound.
Proposition 13. Let G be a graph of order n ≥ 2. If min λ2 (G), λ2 (G) ≤ 1
then
r
n2
λ2 (G) + λ2 (G) ≤ −1 +
− n + 1.
2
Proof. According to [21], the inequality is satisfied for every graph of order
2 ≤ n ≤ 7. Suppose that G is a graph with n ≥ 8 vertices. In this case,
(16)
n
2
+1
2
≤
n2
− n + 1.
2
Let λ2 (G) ≤ 1. From (16) and λ2 (G) ≤ n/2 − 1 [see Theorem 2 of [11]],
λ2 (G) + λ2 (G) ≤
≤
−1 +
n
+1
r2
n2
−1 +
−n+1
2
and, the proof is completed.
Generalized line graphs and exceptional graphs were well studied in [6] and
satisfies λn ≥ −2. The next result shows that such graphs satisfy N G2 -bound.
Proposition 14. Let G be a graph of order n such that λn (G) ≥ −2. Then,
r
λ2 (G) + λ2 (G) ≤ −1 +
n2
− n + 1.
2
Proof. Let G a graph under the conditions above. We have, λ2 (G) ≤ 1 [see Theorem 3 in [5]] and, the result follows directly from Proposition 13.
131
A note on the Nordhaus-Gaddum type inequality to . . .
4. A CLASS OF EXTREMAL GRAPHS TO THE N G2 -BOUND
In this section, we show an infinite family of graphs of girth 3 that satisfies inequality (3). Such graphs were studied by Nikiforov in [17], but here we
will define it using the H-join operation. Let H be a graph with vertex set
V (H) = {vi ; 1 ≤ i ≤ k}. Let F = {Gi ; 1 ≤ i ≤ k} be a family of graphs Gi of
order ni . For each vi ∈ V (H), Gi ∈ F is assigned to vi . The H-join of a graph G,
denoted by G = H[G1 , G2 , . . . , Gk ], in F has vertex set
!
k
[
V (G) =
V (Gi )
i=1
and edge set
E(G) =
k
[

!
E(Gi )
i=1
[

[

{uw : u ∈ V (Gi ), w ∈ V (Gj )} .
vi vj ∈E(H)
If H ' P4 , p, q ≥ 1 and n = 2(p + q), then
Hp,q,q,p = P4 [Kp , Kq , Kq , Kp ].
For n = 2(p + q) + 1,
Hp,q,q,p+1 = P4 [Kp , Kq , Kq , Kp+1 ].
Let us define the following family of graphs:
H(P4 ) = {Hp,q,q,p , Hp,q,q,p+1 ; p, q ≥ 1}.
Figure 1 displays the H−join graph given by H2,5,5,2 ∈ H(P4 ).
Figure 1: H2,5,5,2 ∈ H(P4 ).
It is easy to see that the complement operation is closed in the family H(P4 )
if and only if n is even, that is,
(17)
Hp,q,q,p = P4 (Kp , Kq , Kq , Kp ) = P4 (Kq , Kp , Kp , Kq ) = Hq,p,p,q ∈ H(P4 )
132
N. Abreu, A. E. Brondani, L. de Lima, C. Oliveira
and,
Hp,q,q,p+1 = P4 (Kp , Kq , Kq , Kp+1 ) = P4 (Kq , Kp , Kp+1 , Kq ) ∈
/ H(P4 ).
Let 1 ≤ p ≤ q be integer numbers such that n = 2(p + q) and let r = p − q − 1,
s = q(q + 2p + 2) + (p − 1)2 and t = q(q + 6p − 2) + (p − 1)2 . From Cardoso et
al. [2], the spectrum of Hp,q,q,p is as follows
(18)
Spec(Hp,q,q,p ) =
√
√
√
√ r + t r + 2q + s
r − t r + 2q − s
,
, −1(2p−2) , 0(2q−2) ,
,
.
2
2
2
2
Immediately from (17) and (18), we obtain Proposition 15.
Proposition 15. Let p and q natural numbers such that 1 ≤ p ≤ q. Then,
λ2 (Hp,q,q,p ) = λ2 (Hp,q,q,p ) + q − p.
In the next proposition, we prove that for each n ≡ 0 (mod 4), H n4 , n4 , n4 , n4 is
a P4 -join graph which is extremal to N G2 -bound.
Proposition 16. If G ' Hp,q,q,p , then
λ2 (Hp,q,q,p ) + λ2 (Hp,q,q,p ) = −1 +
p
(q + 6p − 2)q + (p − 1)2 .
Besides, the maximum value for the sum is attained if and only if p =
lnm
q=
.
4
jnk
4
and
Proof. Consider the spectrum of Hp,q,q,p in (18) and Proposition 15. If n = 2(p+q),
p
−16p2 + 8np + (n − 2)2
.
2
√ p
n−α n+α
2
For α = 2 (n − 1) + 1 > 0, define h :
,
→ R such that
4
4
p
−16x2 + 8nx + (n − 2)2
h(x) = −1 +
.
2
n−α n+α
As the function h is continuous in the interval
,
and differ4
4
n−α n+α
n
entiable in
,
, h admits a maximum value at x = . Besides as
4
4
4
jnk
p is a positive number, λ2 (G) + λ2 (G) reaches the maximum value for p =
.
4
lnm
.
Consequentely, q =
4
λ2 (G) + λ2 (G)
=
−1 +
A note on the Nordhaus-Gaddum type inequality to . . .
133
5. CONCLUSION
By Harary [10], if G is a graph of order n ≥ 6 then G or G has girth 3.
In the light of this result, the Propositions 3 and 11 enable us to conclude that
there exist many graphs of girth 3 for which the N G2 -bound is verified. These
facts, in addition to the computational experiments performed for all graphs up to
8 vertices, lead us to conjecture the following result:
Conjecture 17. Let G be a graph on n vertices. Then,
r
n2
− n + 1,
λ2 (G) + λ2 (G) ≤ −1 +
2
with equality if and only if G ' H n4 , n4 , n4 , n4 , for n ≡ 0 (mod 4).
Acknowledgements. The authors would like to thank the anonymous referee
for his/her suggestions that improved the paper. Nair Abreu and Leonardo de
Lima were partially supported by CNPq grants 142241/2014-3 and 309723/2015-9,
respectively.
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∗
Institution: Federal Fluminense University,
Instituto de Cincias Exatas
Departamento de Matemtica,
Desembargador Ellis Hermydio F,
Brazil
E-mail: [email protected]
(Recived 30.11.2016.)
(Revised 22.04.2017.)
A note on the Nordhaus-Gaddum type inequality to . . .
Federal University of Rio de Janeiro,
Brazil,
E-mail: [email protected]
CEFET-RJ,
Av Maracan, 229,
Brazil
E-mail: [email protected]
ENCE,
Brazil
E-mail: [email protected]
135